A recently introduced algorithm for solving the inverse problem for simple classical fluids (i.e. the deduction of the interatomic interaction from structural data), which is based on the fundamental-measure free-energy density functional for hard spheres, is analysed in comparison with other methods. In a benchmark test for the Lennard-Jones system near the triple point, it is comparable with about ten simulations in the iterative predictor - corrector scheme proposed some years ago by Levesque, Weis, and Reatto. The method is used to extract the effective pair potential of Kr from very accurate experimental neutron scattering structure factor data.